MODERN FACTOR ANALYSIS Harry H. Harman «ö THE pigj UNIVERSITY OF CHICAGO PRESS
Contents LIST OF ILLUSTRATIONS GUIDE TO NOTATION xv xvi Parti Foundations of Factor Analysis 1. INTRODUCTION 3 1.1. Brief History of Factor Analysis 3 1.2. Applications of Factor Analysis 6 1.3. Scientific Explanation and Choice 8 2. FACTOR ANALYSIS MODEL 11 2.1. Introduction 11 2.2. Basic Statistics 11 2.3. Linear Model for a Statistical Variable 12 2.4. Variance Components 13 2.5. Factor Patterns and Structures 16 2.6. Factor Patterns as Classical Regression Equations.. 18 2.7. Statistical Fit of the Factor Model 19 2.8. Indeterminateness of Factor Solutions 21 3. MATRIX CONCEPTS ESSENTIAL TO FACTOR ANALYSIS... 24 3.1. Introduction 24 3.2. Basic Concepts of Determinants and Matrices 24 3.3. The Factor Model in Matrix Notation 31 3.4. Solution of Systems of Linear Equations: Method of Substitution 36 3.5. Solution of Systems of Linear Equations: Square Root Method 38 3.6. Calculation of the Inverse of a Matrix 41 4. GEOMETRIC CONCEPTS ESSENTIAL TO FACTOR ANALYSIS 44 4.1. Introduction 44 4.2. Geometry of N Dimensions 44 4.3. Cartesian Coordinate System 46 4.4. Linear Combination and Dependence 47 4.5. Distance Formulas in Rectangular Coordinates.... 52 4.6. Orthogonal Transformations 53 4.7. Angular Separation between Two Lines 55 4.8. Distance and Angle in General Cartesian Coordinates 58 4.9. Geometrie Interpretation of Correlation 60 4.10. Subspaces Employed in Factor Analysis 64 xi
xii 5. THE PROBLEM OF COMMTJNALITY 69 5.1. Introduction 69 5.2. Determination of the Common-Factor Space 69 5.3. Conditions for One Common Factor 73 5.4. Conditions for Two Common Factors 76 5.5. Determination of Communality from Approximate Rank 79 5.6. Numerical Example Employing Approximate Rank 81 5.7. Theoretical Solution for Communality 84 5.8. Arbitrary Approximations to Communality 86 5.9. Complete Approximations to Communality 87 5.10. Examples of Approximations to Communality 91 5.11. Direct Factor Solution 94 6. PROPERTIES OF DIFFERENT TYPES OF FACTOR SOLUTIONS 97 6.1. Introduction 97 6.2. Mathematical and Logical Criteria 99 6.3. Square Root Solutions 102 6.4. Solutions Not Requiring Communalities 103 6.5. Preliminary Solutions Involving Communalities... 109 6.6. Multiple-Factor Solution and Simple Structure Principles 111 6.7. Summary of Factor Solutions 114 Part II Direct Solutions 7. TWO-FACTOR SOLUTION 119 7.1. Introduction 119 7.2. Summation Method 120 7.3. Method of Triads 122 7.4. The Heywood Case 125 8. BI-FACTOR SOLUTION 127 8.1. Introduction 127 8.2. Grouping of Variables 128 8.3. General-Factor Coefficients 131 8.4. Group-Factor Coefficients 132 8.5. Adjustments to the Bi-Factor Solution 133 8.6. Illustrative Examples 135 8.7. Computing Procedures 142 9. PRINCIPAL-FACTOR SOLUTION 154 9.1. Introduction 154 9.2. Derivation of Principal-Factor Method 154 9.3. Additional Theory for Computing Applications 160 9.4. Computing Procedures With Desk Calculator 164
xiii 9.5. Solutions Obtained With Desk Calculators 171 9.6. Outline of Electronic Computer Program 179 9.7. Solutions Obtained with Electronic Computers 185 10. CENTROID SOLUTION 192 10.1. Introduction 192 10.2. Derivation of Centroid Method 192 10.3. Computing Procedures 199 10.4. Illustrative Examples 210 10.5. Averoid Method 211 11. MULTIPLE-GROUP SOLUTION 216 11.1. Introduction 216 11.2. Concepts and Notation 216 11.3. The Oblique Solution 219 11.4. The Orthogonal Solution 222 11.5. Multiple-Group Factor Algorithm 224 11.6. Numerical Illustration 227 Part III Derived Solutions 12. DIFFERENT SOLUTIONS IN COMMON-FACTOR SPACE 233 12.1. Introduction 233 12.2. Relationship between Two Known Solutions 233 12.3. Graphical Procedures for Orthogonal Multiple- Factor Solution 238 12.4. Numerical Illustrations of Orthogonal Multiple- Factor Solutions 245 12.5. Other Problems of Relationships between Factor Solutions 256 13. OBLIQUE MULTIPLE-FACTOR SOLUTIONS 261 13.1. Introduction 261 13.2. Geometrie Basis for an Oblique Solution 262 13.3. Computing Procedures for Oblique Primary-Factor Solution 264 13.4. Oblique Reference Solution 273 13.5. Relationship between Two Types of Oblique Solutions 277 13.6. Numerical Illustrations 280 14. ANALYTICAL METHODS FOR THE MULTIPLE-FACTOR SOLUTION: ORTHOGONAL CASE 289 14.1. Introduction 289 14.2. Rationale for Analytical Methods 290 14.3. Quartimax Method 294 14.4. Varimax Method 301
xiv 15. ANALYTICAL METHODS FOR THE MULTIPLE-FACTOR SOLUTION: OBLIQUE CASE 309 15.1. Introduction 309 15.2. Oblimax Method 309 15.3. Quartimin Method 319 15.4. Oblimin Methods 324 Part IV Special Topics 16. MEASUREMENT OF FACTORS 337 16.1. Introduction 337 16.2. Direct Solution versus Estimation 338 16.3. Complete Estimation Method 338 16.4. Approximation Method 348 16.5. Short Method 349 16.6. Estimation by Minimizing Unique Factors 356 16.7. Factor Measurements by Ideal Variables 360 17. STATISTICAL TESTS OF HYPOTHESES IN FACTOR ANALYSIS 362 17.1. Introduction 362 17.2. Statistical Estimation 364 17.3. Maximum-Likelihood Estimates of Factor Loadings 366 17.4. Test of Significance for the Number of Common Factors 370 17.5. Computing Procedures 372 17.6. Numerical Illustrations 378 17.7. Concluding Remarks 380 Part V Problems and Exercises PROBLEMS 387 ANSWERS 411 Appendix STATISTICAL TABLES 439 BIBLIOGRAPHY 445 INDEX 465